1. Spatial Data Mining

2. Outline Motivation, Spatial Pattern Families
Limitations of Traditional Statistics
Colocations and Co-occurrences
Spatial outliers
Summary: What is special about mining spatial data?

3. Why Data Mining? Holy Grail - Informed Decision Making
Sensors & Databases increased rate of Data Collection
Transactions, Web logs, GPS-track, Remote sensing, …
Challenges:
Volume (data) >> number of human analysts
Some automation needed
Approaches
Database Querying, e.g., SQL3/OGIS
Data Mining for Patterns …

4. Data Mining vs. Database Querying
Recall Database Querying (e.g., SQL3/OGIS)
Can not answer questions about items not in the database!
Ex. Predict tomorrow’s weather or credit-worthiness of a new customer
Can not efficiently answer complex questions beyond joins
Ex. What are natural groups of customers?
Ex. Which subsets of items are bought together?
Data Mining may help with above questions!
Prediction Models
Clustering, Associations, …

5. Spatial Data Mining (SDM)
The process of discovering
interesting, useful, non-trivial patterns from large spatial datasets
Spatial pattern families
Hotspots, Spatial clusters
Spatial outlier, discontinuities
Co-locations, co-occurrences
Location prediction models …

6. Pattern Family 1: Co-locations/Co-occurrence
Given: A collection of different types of spatial events
Find: Co-located subsets of event types
Source: Discovering Spatial Co-location Patterns: A General Approach, IEEE Transactions
on Knowledge and Data Eng., 16(12), December 2004 (w/ H.Yan, H.Xiong).

7. Pattern Family 2: Hotspots, Spatial Cluster
The 1854 Asiatic Cholera in London
Near Broad St. water pump except a brewery

8. Complicated Hotspots Complication Dimensions Challenges: Trade-off b/w
Time
Spatial Networks
Challenges: Trade-off b/w
Semantic richness and
Scalable algorithms

9. Pattern Family 3: Predictive Models
Location Prediction:
Predict Bird Habitat Prediction
Using environmental variables
E.g., distance to open water
Vegetation durability etc.

10. Pattern Family 4: Spatial Outliers
Spatial Outliers, Anomalies, Discontinuities
Traffic Data in Twin Cities
Abnormal Sensor Detections
Spatial and Temporal Outliers
Source: A Unified Approach to Detecting Spatial Outliers, GeoInformatica, 7(2), Springer, June 2003.
(A Summary in Proc. ACM SIGKDD 2001) with C.-T. Lu, P. Zhang.

11. What’s NOT Spatial Data Mining (SDM)
Simple Querying of Spatial Data
Find neighbors of Canada, or shortest path from Boston to Houston
Testing a hypothesis via a primary data analysis
Ex. Is cancer rate inside Hinkley, CA higher than outside ?
SDM: Which places have significantly higher cancer rates?
Uninteresting, obvious or well-known patterns
Ex. (Warmer winter in St. Paul, MN) => (warmer winter in Minneapolis, MN)
SDM: (Pacific warming, e.g. El Nino) => (warmer winter in Minneapolis, MN)
Non-spatial data or pattern
Ex. Diaper and beer sales are correlated
SDM: Diaper and beer sales are correlated in blue-collar areas (weekday evening)

12. Quiz
Categorize following into queries, hotspots, spatial outlier, colocation, location prediction:
(a) Which countries are very different from their neighbors?
(b) Which highway-stretches have abnormally high accident rates ?
(c) Forecast landfall location for a Hurricane brewing over an ocean?
(d) Which retail-store-types often co-locate in shopping malls?
(e) What is the distance between Beijing and Chicago?

13. Limitations of Traditional Statistics
Classical Statistics
Data samples: independent and identically distributed (i.i.d)
Simplifies mathematics underlying statistical methods, e.g., Linear Regression
Certain amount of “clustering” of spatial events
Spatial data samples are not independent
Spatial Autocorrelation metrics
Global and local Moran’s I
Spatial Heterogeneity
Spatial data samples may not be identically distributed!
No two places on Earth are exactly alike!

14. “Degree of Clustering”: K-Function
Purpose: Compare a point dataset with a complete spatial random (CSR) data
Input: A set of points
where λ is intensity of event
Interpretation: Compare k(h, data) with K(h, CSR)
K(h, data) = k(h, CSR): Points are CSR
> means Points are clustered
< means Points are de-clustered
[number of events within distance h of an arbitrary event]
CSR
Clustered
De-clustered

15. Cross K-Function [number of type j event within distance h
Cross K-Function Definition
Cross K-function of some pair of spatial feature types
Example
Which pairs are frequently co-located
Statistical significance
[number of type j event within distance h
of a randomly chosen type i event]

16. Estimating K-Function
[number of events within distance h of an arbitrary event]
𝐾 ℎ = 𝜆 −1 ∗ 𝑖 𝑗≠𝑖 𝐼 𝑑 𝑖𝑗 <ℎ ; 𝜆= #𝑒𝑣𝑒𝑛𝑡𝑠 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑡𝑢𝑑𝑦 𝑟𝑒𝑔𝑖𝑜𝑛
[number of type j event within distance h
of a randomly chosen type i event]
𝐾 𝑖𝑗 ℎ = ( 𝜆 𝑖 𝜆 𝑗 𝐴) −1 ∗ 𝑘 𝑙 𝐼( 𝑑 𝑖 𝑘 𝑗 𝑙 <ℎ) ; A is the area of the study region.

17. Recall Pattern Family 2: Co-locations
Given: A collection of different types of spatial events
Find: Co-located subsets of event types
Source: Discovering Spatial Co-location Patterns: A General Approach, IEEE Transactions
on Knowledge and Data Eng., 16(12), December 2004 (w/ H.Yan, H.Xiong).

18. Illustration of Cross-Correlation
Illustration of Cross K-function for Example Data
Cross-K Function for Example Data

19. Background: Association Rules
Association rule e.g. (Diaper in T => Beer in T)
Support: probability (Diaper and Beer in T) = 2/5
Confidence: probability (Beer in T | Diaper in T) = 2/2
Apriori Algorithm
Support based pruning using monotonicity

20. Apriori Algorithm
How to eliminate infrequent item-sets as soon as possible?
Support threshold >= 0.5

21. Apriori Algorithm Eliminate infrequent singleton sets
Support threshold >= 0.5
Milk
Cookies
Bread
Eggs
Juice
Coffee

22. Apriori Algorithm
Make pairs from frequent items & prune infrequent pairs!
Item type
Count
Milk, Juice 2
Bread, Cookies
Milk, cookies 1
Milk, bread
Bread, Juice
Cookies, Juice
Support threshold >= 0.5
Milk
Cookies
Bread
Eggs
Juice
Coffee MB BJ MJ BC MC CJ 81

23. Apriori Algorithm
Make triples from frequent pairs & Prune infrequent triples!
Item type
Count
Milk, Juice 2
Bread, Cookies
Milk, Cookies 1
Milk, bread
Bread, Juice
Cookies, Juice
Support threshold >= 0.5
Milk
Cookies
Bread
Eggs
Juice
Coffee
MBCJ
MBC
MBJ
MCJ
BCJ
No triples generated due to monotonicity!
How?? MB MC MJ BC BJ CJ
Apriori algorithm
examined only
12 subsets
instead of 64!

24. Association Rules Limitations
Transaction is a core concept!
Support is defined using transactions
Apriori algorithm uses transaction based Support for pruning
However, spatial data is embedded in continuous space
Transactionizing continuous space is non-trivial !

25. Spatial Association (Han 95) vs. Cross-K Function
Spatial Association Rule (Han 95)
Output = (B,C) with threshold 0.5
Transactions by Reference feature, e.g. C
Transactions: (C1, B1), (C2, B2)
Support (A,B) = Ǿ
Support(B,C)=2 / 2 = 1
Input = Feature A,B, and, C, & instances A1, A2, B1, B2, C1, C2

26. Spatial Association (Han 95) vs. Cross-K Function
Spatial Association Rule (Han 95)
Output = (B,C) with threshold 0.5
Transactions by Reference feature, e.g. C
Transactions: (C1, B1), (C2, B2)
Support (A,B) = Ǿ
Support(B,C)=2 / 2 = 1
Cross-K Function
Cross-K (A, B) = 2/4 * (area)
Cross-K(B, C) = 2/4 * (area)
Cross-K(A, C) = 0
Input = Feature A,B, and, C, & instances A1, A2, B1, B2, C1, C2
Output = (A,B), (B, C) with appropriate threshold

27. Spatial Colocation (Shekhar 2001)
Features: A. B. C
Feature Instances: A1, A2, B1, B2, C1, C2
Feature Subsets: (A,B), (A,C), (B,C), (A,B,C)
Participation ratio (pr):
pr(A, (A,B)) = fraction of A instances neighboring feature {B} = 2/2 = 1
pr(B, (A,B)) = ½ = 0.5
Participation index (A,B) = pi(A,B) = min{ pr(A, (A,B)), pr(B, (A,B)) } = min (1, ½ ) = 0.5
pi(B, C) = min{ pr(B, (B,C)), pr(C, (B,C)) } = min (1,1) = 1
Participation Index Properties:
(1) Computational: Non-monotonically decreasing like support measure
(2) Statistical: Upper bound on Ripley’s Cross-K function

28. Participation Index >= Cross-K Function
B.1
A.2
B.2
Cross-K (A,B)
2/6 = 0.33
3/6 = 0.5
6/6 = 1
PI (A,B)
2/3 = 0.66 1

29. Association Vs. Colocation
Associations
Colocations
underlying space
Discrete market baskets
Continuous geography
event-types
item-types, e.g., Beer
Boolean spatial event-types
collections
Transaction (T)
Neighborhood N(L) of location L
prevalence measure
Support, e.g., Pr.[ Beer in T]
Participation index, a lower bound on Pr.[ A in N(L) | B at L ]
conditional probability measure
Pr.[ Beer in T | Diaper in T ]
Participation Ratio(A, (A,B)) =
Pr.[ A in N(L) | B at L ]

30. Spatial Association Rule vs. Colocation
Spatial Association Rule (Han 95)
Output = (B,C)
Transactions by Reference feature C
Transactions: (C1, B1), (C2, B2)
Support (A,B) = Ǿ, Support(B,C)=2 / 2 = 1
Cross-K Function
Cross-K (A, B) = 2/4 * (area)
Cross-K(B, C) = 2/4 * (area)
Output = (A,B), (B, C)
Input = Spatial feature A,B, C, & their instances
Colocation - Neighborhood graph
Output = (A,B), (B, C)
PI(A,B) = min(2/2,1/2) = 0.5
PI(B,C) = min(2/2,2/2) = 1

31. Mining Colocations: Problem Definition
Input:
(a) K Boolean feature types (e.g, presence of a bird nest, tree, etc.)
(b) Feature Instances <feature type, location>
(c) A neighbor relation R over the locations
(d) Prevalence_threshold 𝜃 (threshold on participation-index)
Output:
(a) All Co-location rules with prevalence > 𝜃
Co-location rules are defined as subsets of feature-types
Each instance of co-location rules is a “neighborhood”

32. Mining Colocations: Basic Concepts

33. Key Concepts: Neigborhood

34. Key Concepts: Co-location rules

35. Some more Key Concepts

36. Mining Colocations: Algorithm Trace

37. Mining Colocations: Algorithm Trace (1/6)

38. Mining Colocations: Algorithm Trace (2/6)

39. Mining Colocations: Algorithm Trace (3/6)

40. Mining Colocations: Algorithm Trace (5/6)

41. Mining Colocations: Algorithm Trace (6/6)

42. Quiz Which is false about concepts underlying association rules?
a) Apriori algorithm is used for pruning infrequent item-sets
b) Support(diaper, beer) cannot exceed support(diaper)
c) Transactions are not natural for spatial data due to continuity of
geographic space
d) Support(diaper) cannot exceed support(diaper, beer)

43. Outliers: Global (G) vs. Spatial (S)

44. Outlier Detection Tests: Variogram Cloud
Graphical Test: Variogram Cloud

45. Outlier Detection Test: Moran Scatterplot
Graphical Test: Moran Scatter Plot

46. Neighbor Relationship: W Matrix

47. Outlier Detection – Scatterplot
Quantitative Tests: Scatter Plot

48. Outlier Detection Tests: Spatial Z-test
Quantitative Tests: Spatial Z-test
Algorithmic Structure: Spatial Join on neighbor relation

49. Quiz Which of the following is false about spatial outliers?
a) Oasis (isolated area of vegetation) is a spatial outlier area in a desert
b) They may detect discontinuities and abrupt changes
c) They are significantly different from their spatial neighbors
d) They are significantly different from entire population

50. Statistically Significant Clusters
K-Means does not test Statistical Significance
Finds chance clusters in complete spatial randomness (CSR)
Classical
Clustering
Spatial

51. Spatial Scan Statistics (SatScan)
Goal: Omit chance clusters
Ideas: Likelihood Ratio, Statistical Significance
Steps
Enumerate candidate zones & choose zone X with highest likelihood ratio (LR)
LR(X) = p(H1|data) / p(H0|data)
H0: points in zone X show complete spatial randomness (CSR)
H1: points in zone X are clustered
If LR(Z) >> 1 then test statistical significance
Check how often is LR( CSR ) > LR(Z)
using 1000 Monte Carlo simulations

52. SatScan Examples Test 1: Complete Spatial Randomness
SatScan Output: No hotspots !
Highest LR circle is a chance cluster!
p-value = 0.128
Test 2: Data with a hotspot
SatScan Output: One significant hotspot!
p-value = (low p-value is good)

53. Location Prediction Problem
Target Variable: Nest Locations
Vegetation Index
Water Depth
Distance to Open Water

54. Location Prediction Models
Traditional Models, e.g., Regression (with Logit or Probit),
Bayes Classifier, …
Spatial Models
Spatial autoregressive model (SAR)
Markov random field (MRF) based Bayesian Classifier